Dynamic programming and the Lagrange multipliers (Q749458)

Theoretically, every optimization problem with constraints can be solved by the method of Lagrange multipliers or the dynamic programming technique. This paper considers problem (1): Let \(a_ 1\leq a_ 2\leq...\leq a_ n\) be nonnegative reals (n\(\geq 2)\) such that \[ \sum^{n}_{j=1}a_ ja_{+1}=1\quad (a_{n+1}=a_ 1). \] Determine the minimum value of \(\sum^{n}_{j=1}a_ j\). As a result, the author not only demonstrated that the dynamic programming technique is far superior to the method of Lagrange multipliers in solving the problem (1) but also revealed that the dynamic programming technique is used to solve a problem by tackling a sequence of its special cases so that one gradually unfolds the hidden complexities of the problem and eventually finds more basic and simpler means of solving the problem.